Available via license: CC BY 4.0
Content may be subject to copyright.
Understanding the interplay of entanglement and nonlocality:
motivating and developing a new branch of entanglement theory
David Schmid1,2,3, Thomas C. Fraser1,2, Ravi Kunjwal4, Ana Belén Sainz3, Elie Wolfe1, and
Robert W. Spekkens1
1Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, Ontario Canada N2L 2Y5
2Institute for Quantum Computing and Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario N2L 3G1,
Canada
3International Centre for Theory of Quantum Technologies, University of Gdańsk, 80-308 Gdańsk, Poland
4
Centre for Quantum Information and Communication, Ecole polytechnique de Bruxelles, CP 165, Université libre de Bruxelles, 1050
Brussels, Belgium
A standard approach to quantifying
resources is to determine which operations
on the resources are freely available, and to
deduce the partial order over resources that
is induced by the relation of convertibility
under the free operations. If the resource
of interest is the nonclassicality of the
correlations embodied in a quantum state,
i.e., entanglement, then the common
assumption is that the appropriate choice
of free operations is Local Operations and
Classical Communication (LOCC). We
here advocate for the study of a different
choice of free operations, namely, Local
Operations and Shared Randomness (LOSR),
and demonstrate its utility in understanding
the interplay between the entanglement of
states and the nonlocality of the correlations
in Bell experiments. Specifically, we show
that the LOSR paradigm (i) provides a
resolution of the anomalies of nonlocality,
wherein partially entangled states exhibit
more nonlocality than maximally entangled
states, (ii) entails new notions of genuine
multipartite entanglement and nonlocality
that are free of the pathological features of the
conventional notions, and (iii) makes possible
a resource-theoretic account of the self-testing
of entangled states which generalizes and
simplifies prior results. Along the way, we
derive some fundamental results concerning
the necessary and sufficient conditions for
convertibility between pure entangled states
under LOSR and highlight some of their
consequences, such as the impossibility of
catalysis for bipartite pure states. The
resource-theoretic perspective also clarifies
why it is neither surprising nor problematic
that there are mixed entangled states which
do not violate any Bell inequality. Our results
motivate the study of LOSR-entanglement as
a new branch of entanglement theory.
Contents
1 Introduction 1
2 Nonclassicality of correlations for states
and boxes 4
3 Resolving the anomaly of nonlocality 6
4 Genuine multipartite entanglement 9
5 Self-testing of entangled states 12
6 The resource theory of LOSR-entanglement 17
7 The ordering of entangled states depends
on the network structure 20
8 On entangled states that do not violate any
Bell inequality 23
9 Discussion 25
Acknowledgments 27
References 27
Appendices 32
1 Introduction
The term “entangled” was first used only for pure
states of a composite system, and meant simply
Accepted in Quantum 2023-11-27, click title to verify. Published under CC-BY 4.0. 1
arXiv:2004.09194v4 [quant-ph] 30 Nov 2023
that the state was not a tensor product of states
of the components [1]. Thus, for pure states,
entanglement is synonymous with correlation.
When the quantum information community turned
its attention to mixed states, the term “entangled”
obtained a broader meaning, aimed at capturing
the nonclassicality of correlations. Specifically, a
quantum state was taken to exhibit nonclassical
correlations if it could not be expressed as a
mixture of product states [2], in which case
it was called nonseparable. Shortly thereafter, it
was realized that entangled states (both pure
and mixed) could be used to implement useful
information-processing tasks, and they began to
be studied as a resource. Because the tasks being
considered at the time mainly concerned the
resourcefulness of entangled states in circumstances
wherein the separated parties had access to
classical communication channels (for instance,
their use in simulating quantum channels via
the teleportation protocol [3], and in enhancing
communication via the dense coding protocol [4]), it
was natural to define the interconvertibility preorder
of entangled states relative to Local Operations
and Classical Communication (LOCC) [5]. This
choice was consistent with the previous definition of
the boundary between entangled and unentangled
states, since the states one can prepare freely by
LOCC are precisely the separable states.
However, LOCC is not the only choice of free
operations that could have been used to formalize
the notion of entanglement as a resource. Consider
the set of Local Operations and Shared Randomness
(LOSR), wherein the parties have access to a
common source of classical randomness, but no
classical channels among them. If one chooses LOSR
as the set of free operations, one also reproduces
the standard definition of entangled states as
nonseparable states, since the free states relative
to LOSR are also the separable ones. The ordering
induced over entangled states by LOSR, however, is
different from the one induced by LOCC, even in the
case of pure states, as we will show. Consequently,
quantification of entanglement relative to LOSR
leads to quite different results than one obtains by
quantifying it relative to LOCC.
To distinguish these two notions of entanglement,
we will henceforth use the terms LOCC-
entanglement and LOSR-entanglement.
In this article, we advocate for the development
of the theory of LOSR-entanglement. We motivate
its study by demonstrating how much light it sheds
on the interplay of entanglement and nonlocality.
Specifically, we argue that for one of the most
natural ways of conceptualizing a Bell scenario, it
is LOSR-entanglement that is the relevant resource
of entanglement, rather than LOCC-entanglement.
We describe many ways in which conceptual
puzzles regarding the interplay of entanglement and
nonlocality are resolved in this approach. The notion
of LOSR-entanglement was originally proposed by
Buscemi [6], also in the context of Bell scenarios,
but no further work has been done to date on
characterizing it. We hope that the arguments
provided herein for its importance will motivate
researchers to turn their attention to it.
The term ‘box’ will here be used as jargon for
a multipartite process with only classical inputs
and classical outputs which can be realized by
a common source (either classical or quantum)
which is shared among the parties and subjected
to local measurements. In other words, a box has
the structure of a Bell experiment.1Formally, a
box is represented by the conditional probability
distribution over its classical outputs given its
classical inputs. Boxes can be divided into those
whose correlational properties are classical and
those for which they exhibit nonclassicality, where
the division is based on whether the corresponding
conditional probability distribution satisfies all the
Bell inequalities or not. These two classes are
conventionally termed “local” and “nonlocal”. 2
1
Such processes are termed ‘common-cause boxes’ in
Ref. [
7
]. Note that we are here only interested in boxes that
are quantumly realizable, rather than the strictly larger set
of boxes that are realizable in the framework of Generalized
Probabilistic Theories [
8
,
9
]. Note also that the term ‘box’
is sometimes used in a manner that does not presume that
the internal causal structure is that of local measurements
on a common source. This is done, for instance, by authors
who would prefer to make no assumptions about a box’s inner
workings and to rely instead on assumptions about the spatio-
temporal relations among its inputs and outputs. We discuss
this alternative approach in Sec. 7.
2
Although we are following a standard convention in
referring to such nonclassicality of boxes as “nonlocality”, we
note that this is merely for the sake of making our article easier
to read. The conventional terminology is actually a potential
source of confusion insofar as it suggests a commitment to
a view that many (including the present authors) do not
endorse, namely, that the correct explanation of Bell inequality
violations involves superluminal causes. See Sec. 2.3.1 of
Accepted in Quantum 2023-11-27, click title to verify. Published under CC-BY 4.0. 2
Many authors have argued that entanglement and
nonlocality are simply different kinds of resources.
Indeed, this is a standard response to some of the
puzzling features of their interplay. In this article,
however, we take a different point of view. We
argue that entanglement and nonlocality quantify
the same notion of resourcefulness for the processes
to which they apply, namely, the nonclassicality
of the correlational properties of those processes.
Entanglement refers to the nonclassicality of the
correlational properties of quantum states, while
nonlocality refers to the nonclassicality of the
correlational properties of boxes.
Furthermore, we argue that whether given
correlational properties (of a state or of a box) should
be deemed nonclassical depends on the network
connecting the parties.
Prior results that appeared puzzling are seen, in
retrospect, to be a result of mixing together notions
of nonclassicality related to different networks. We
demonstrate that the choice of network structure
that fits best with pre-existing ideas regarding the
interplay of the nonclassicality of correlations of
states (entanglement) and the nonclassicality of
correlations of boxes (nonlocality) is the network
where the parties merely share a common source. In
such a network, the relevant notion of entanglement
is LOSR-entanglement.
We now summarize the rest of the article.
In Section 2, we explain why a resource
theory that encompasses both entangled states
and nonlocal boxes—as different types of resources
of nonclassicality of correlations—must be based
on a type-independent constraint defining the
free operations, which is then particularized to
conversion relations among specific types, such as
conversions from states and boxes. (Note that one
has no choice but to work within such a mixed-type
resource theory, because conversions from entangled
states to nonlocal boxes are precisely the focus of
any study of the interplay of entanglement and
Ref. [
7
] for more discussion of this issue. Note, furthermore,
that the adjective ‘nonlocal’ has sometimes been used to
delineate those quantum states that can be used to violate a
Bell inequality in the Bell scenario. As we argue in Appendix 8,
however, being nonlocal in this sense should not be considered a
necessary condition for the correlation properties of a quantum
state to be judged nonclassical. In any case, in this article, we
will use the term “nonlocal” solely as a descriptor of boxes,
where we will take itto signify nonclassicality of the correlations
that the box describes.
nonlocality in Bell scenarios.) We explain why these
free operations must include all of LOSR if the
objective is to quantify the nonclassicality of the
correlations.
The three following sections of the article
explain how a reconsideration of Bell scenarios in
terms of LOSR-entanglement (rather than LOCC-
entanglement) resolves some problems and clarifies
many issues from the Bell literature.
In Section 3, we consider various conceptual
puzzles surrounding anomalies of nonlocality [10–
25], that is, situations wherein features of nonlocal
boxes are found to be realizable by a partially
entangled state but not by a maximally entangled
state. The lesson that has until now been drawn from
these anomalies is that, in spite of prior intuitions to
the contrary, there are measures of the nonlocal yield
of a state (i.e., the nonlocality of boxes that can be
obtained from the state) that are not monotonically
related to the state’s entanglement. We show,
however, that there is a more productive conclusion
to be drawn, namely, that the counterintuitive
features of the anomalies are best understood to
be a consequence of implicitly evaluating state to
box conversions relative to LOSR but state to
state conversions (and thus entanglement) relative
to LOCC. The interplay of entanglement and
nonlocality becomes intuitive if one instead takes the
appropriate notion of entanglement to be the one
based on LOSR. One’s prior intuitions are in fact
vindicated when one proceeds in this fashion. For
instance, we show that every measure of the nonlocal
yield of a given state is a valid measure of the state’s
LOSR-entanglement.
In Section 4, we show that by focussing on LOSR-
entanglement rather than LOCC-entanglement, one
can resolve an analogous (but not previously
articulated) anomaly concerning the interconversion
between genuine 3-way entangled states and genuine
3-way nonlocal boxes. The resolution highlights
the fact that the notion of genuine multipartite
entanglement changes when entanglement is judged
relative to LOSR rather than LOCC. Furthermore,
our notion of genuine multipartite entanglement
does not have a pathological property that the
traditional notion exhibits, namely, the failure of the
closure under tensor products of the states which are
not genuinely multipartite entangled [26–28].
In Section 5, we demonstrate that well-
known results concerning self-testing of entangled
Accepted in Quantum 2023-11-27, click title to verify. Published under CC-BY 4.0. 3
states [29–32] can be better understood in terms
of the interplay of the nonlocality of boxes and the
LOSR-entanglement of states. While our resource-
theoretic approach to self-testing coincides with
the usual approach for pure states and convexly
extremal boxes, we show that it provides a corrective
to the standard definition for the case of mixed
states and convexly nonextremal boxes. In doing
so, we show that both of these appear in nontrivial
instances of self-testing, despite previous claims to
the contrary. We also show that the clarity of our
principled approach to self-testing makes it easy to
resolve ambiguous cases (e.g., chiral states), and to
extend self-testing to novel scenarios and even novel
types of resources and novel resource theories.
In Section 6, we derive some general results about
the preorder of pure entangled states under LOSR.
These results will justify some of the critical steps in
our arguments, so we will be referencing forward to
them throughout the text.
Section 7explains why the nonclassicality of
correlations, and thus entanglement, is only defined
relative to a network structure among the parties.
We also note that one can study the interplay of
entanglement and nonlocality in a network structure
incorporating classical communication between the
parties by restricting attention to boxes that have
space-like separated wings. We contrast the nature
of this interplay with the one observed for the
network with common sources, and we highlight
what needs to be done to properly formalize such an
approach.
In Section 8, we discuss the consequences of
our approach for what is often taken to be a
surprising aspect of the interplay of entanglement
and nonlocality, namely, the fact that there are
mixed entangled states that cannot violate any Bell
inequality.
Finally, in Section 9, we provide a discussion of
the results and future work.
2
Nonclassicality of correlations for states
and boxes
Understanding the interplay between the
entanglement of states and the nonlocality of boxes
means understanding whether particular types and
measures of entanglement of states are required to
realize particular types and measures of nonlocality
of boxes. In order to do so, one must articulate
precisely what operations are assumed to be freely
available in converting states to boxes. But in
addition to this, one must specify what operations
are freely available in achieving conversions among
boxes, because the convertibility relations among
boxes determine measures of nonlocality (via order-
preserving functions, i.e., monotones), and one
must also specify what operations can be used
to achieve conversions among states, because the
convertibility relations among states determine
measures of entanglement. Consequently, there are
three choices of free operations of interest—those
governing box-to-box conversions, those governing
state-to-box conversions, and those governing state-
to-state conversions.
The free operations governing each of these type-
specific varieties of conversion cannot be stipulated
arbitrarily. They must be understood as being
induced by some type-independent constraint that
is then particularized to these cases. It has been
argued elsewhere [33,34] that a given choice of
the set of free operations in a resource theory
is physically interesting (as opposed to being of
mere mathematical interest) only if it is motivated
by some restriction on physical or experimental
capabilities.3Insofar as a preparation of a resource
of a given type is also a kind of conversion relation,
namely, from the trivial type (no systems) to the
type of the resource, the boundary between free
and nonfree for every different type of resource
also cannot be stipulated arbitrarily but is induced
by the type-independent constraint that is then
particularized.
Historically, the question of whether a given
entangled state can generate a given nonlocal box
has been interpreted as the question of whether
there exists some set of quantum measurements
on each wing that can be implemented on the
entangled state to yield the conditional probability
distribution of outcomes given settings which is
associated to the nonlocal box. (For instance, this
is the case in discussions of self-testing of states by
boxes, as we note in Sec. 5.) In other words, whether a
given state-to-box conversion relation holds or not is
3
In the framework for resource theories set up in Ref. [
33
],
the nature of the physical restriction is presumed to have some
structural properties, such as the free operations being closed
under parallel and serial composition.
Accepted in Quantum 2023-11-27, click title to verify. Published under CC-BY 4.0. 4
traditionally evaluated relative to Local Operations
(LO). The reason, presumably, is that state-to-box
conversions have heretofore been conceptualized as
analogues of a Bell experiment, wherein the choice
of local measurement at one wing has traditionally
been presumed to be independent of the choice at
any other wing, even though this independence is
not needed to derive the Bell inequalities.4
When one conceptualizes state-to-box
conversions in a resource-theoretic way, however, it
becomes apparent that this LO-based approach is
untenable, as we now demonstrate.
First recall that, as we noted in the introduction,
the entanglement of states and the nonlocality
of boxes quantify the nonclassicality of the
correlational properties of states and boxes. For
both states and boxes, the distinction between free
and nonfree is the distinction between classical and
nonclassical correlations. In the case of quantum
states, this corresponds to the distinction between
separable and nonseparable, while in the case of
boxes, it corresponds to the distinction between
satisfying all Bell inequalities and violating some
Bell inequality. Next, note that the set of separable
states and the set of Bell-inequality-satisfying boxes
cannot be generated by local operations alone; they
require shared randomness.
Because a preparation of a box is a special
case of a state-to-box conversion where the
input type is trivial, if one were to assume LO
as the set of free operations for state-to-box
conversions, one would be stipulating that the
distinction between free and nonfree boxes is the
distinction between uncorrelated and correlated
(i.e., product and nonproduct forms), rather than
the distinction between Bell-inequality-satisfying
and Bell-inequality-violating. Consequently, an
LO-based approach cannot capture the classical-
nonclassical distinction.
Furthermore, if the free operations are to be
independent of type, then if one were to take LO
as the set of free operations governing state-to-box
conversions, one would also have to take LO to
also govern state-to-state conversions, so that the
distinction between free and nonfree states would
also correspond to the distinction between product
4
The assumption that the setting variables are independent
of the hidden variables, on the other hand, is needed to derive
the Bell inequalities.
and nonproduct forms, rather than the distinction
between separable and nonseparable, and thus
would again not capture the classical-nonclassical
distinction. One must therefore reject the historical
LO-based approach to the study of state-to-box
conversions.
We now articulate our preferred approach to
a resource-theoretic study of the interplay of
entanglement and nonlocality. We assume that the
parties are connected by a network wherein they
all have access to a common source, but where
there are no channels between them, so that the
distinction between free and nonfree operations is
the distinction between what can be achieved by a
common classical source (shared randomness) and
what can be achieved by a common quantum source
(nonseparable states). This is a type-independent
restriction. It follows that the set of free operations
governing all varieties of conversion relations,
including state-to-state, state-to-box, and box-to-
box, is LOSR. Further discussion of this proposal is
provided in Sec. 7.
An alternative approach is one wherein the
network includes channels among the parties,
implying that LOCC is the set of free operations
for all varieties of resource conversion. At first
glance, it might seem that the latter approach
cannot possibly capture the nonlocality of boxes,
as classical communication can be used to simulate
any Bell inequality violation without requiring
nonclassicality. As we note in Sec. 7, however, such
a conceptualization can be made consistent by
restricting attention to a subclass of boxes, and it
may be the pertinent one for certain applications.
Nonetheless, we shall show in Sec. 7that it is rather
more difficult to formalize than the one we pursue
here and that the interplay of entanglement and
nonlocality that it implies involves a more significant
departure from standard intuitions than the one
based on LOSR. This also motivates our focus on
LOSR in this article.
As we argued above, one cannot leave out shared
randomness when assessing resource conversions if
the resource of interest is the nonclassicality of
states and boxes. In spite of this, there are special
cases of state-to-box conversions wherein the shared
randomness does not offer any additional power over
LO. This occurs if the box is convexly extremal in
Accepted in Quantum 2023-11-27, click title to verify. Published under CC-BY 4.0. 5
the set of quantumly realizable boxes.5Similarly,
for box-to-box conversions where the output box is
convexly extremal, LO and LOSR also deliver the
same verdicts about convertibility relations. In fact,
there is a slightly larger set of output boxes for which
LO and LOSR deliver the same verdicts for state-to-
box and box-to-box conversions, namely, those that
are LO-equivalent to a convexly extremal box. The
result can be summarized as follows:
Lemma 1. Consider the following statements about
interconversion between an
n
-partite state
ρ
and an
n-partite box B
(i) ρ7→Bby LOSR,
(ii) ρ7→Bby LO,
and between a pair of n-partite boxes, B0and B,
(i)′B07→Bby LOSR,
(ii)′B07→Bby LO.
The following implications hold among these
conditions:
(a)
If
B
is a convexly extremal box or a convexly
nonextremal box that is LO-equivalent to a
convexly extremal box, then (i) and (ii) are
equivalent and (i)′and (ii)′are equivalent.
(b)
If
B
is an arbitrary convexly nonextremal box
then although it is still the case that (ii) =
⇒
(i)
and (ii)
′
=
⇒
(i)
′
, it can happen that (i)
=⇒
(ii) and it can happen that (i)′=⇒(ii)′.
Here, convex extremality is judged relative to the set
of quantumly realizable boxes
Proof.
For all boxes
B
, (ii) =
⇒
(i) and (ii)
′
=
⇒
(i)
′
because LO is a strict subset of LOSR. It therefore
suffices to consider only the reverse implications.
Claim (a). That (i) =
⇒
(ii) (respectively, (i)
′
=
⇒
(ii)
′
) for a convexly extremal
B
is seen as
follows: if a mixture of different LO operations
takes
ρ
(respectively
B0
) to
B
, then by the convex-
extremality of
B
, every LO operation in the mixture
must take
ρ
(respectively
B0
) to
B
. Now consider
the case where
B
is convexly nonextremal but LO-
equivalent to a convexly extremal box, which we
5
Here, a box is said to be quantumly realizable if it can be
obtained from some quantum state by some LOSR operation.
Note, however, that one could equally well define a box to be
quantum realizable if it can be obtained from a quantum state
by an LO operation, since the shared randomness can always
be provided by the quantum state.
denote by
Bext
. Note that
Bext
is also LOSR-
equivalent to
B
, since LOSR subsumes LO. By
assumption,
ρ7→B
by LOSR (respectively
B07→B
by
LOSR). It then follows from the LOSR-equivalence
of
B
and
Bext
that
ρ7→Bext
by LOSR (respectively
B07→ Bext
by LOSR). Next, from the fact that
(i) =
⇒
(ii) for convexly extremal boxes, it follows
that
ρ7→ Bext
by LO (respectively
B07→ Bext
by
LO). Finally, given the LO-equivalence of
B
and
Bext
, we obtain
ρ7→B
by LO (respectively
B07→B
by LO). Claim (b). To see that there are convexly
nonextremal boxes
B
for which (i)
′=⇒
(ii)
′
, it
suffices to consider the case where
B0
is a product
box while
B
=
B0⊗Bextra
where
Bextra
is any local
box that is not a product box, so that it can be
prepared for free using LOSR operations, but not
using LO operations. (Here ‘
⊗
’ denotes parallel
composition of resources [
33
].) To see that there are
convexly nonextremal boxes
B
for which (i)
=⇒
(ii),
we can take
B
=
B′⊗Bextra
where
B′
is any convexly
extremal box that self-tests ρ.
Because of this lemma, some pre-existing results
concerning state-to-box conversions under LO
coincide with results about state-to-box conversions
under LOSR. In such cases, the LO-based
assessments of which state-to-box conversions are
possible coincide with the LOSR-based assessments
and one can simply incorporate all previous results
based on LO into a resource theory based on
LOSR. We will see that considerations of state-to-
box conversions in discussions of the anomaly of
nonlocality, studied in Sec. 3, are of this sort. Other
pre-existing results, however, do need to b e corrected
if one is to understand them as results in a resource
theory based on LOSR. The precise conditions for
the self-testing of states by boxes, considered in
Sec. 5, are of this sort.
3 Resolving the anomaly of nonlocality
As summarized in the introduction and in Ref. [10],
the anomaly of nonlocality refers to the fact that
there are situations wherein features of nonlocal
boxes are found to be realizable by a partially
entangled state but not by a maximally entangled
state. The recognition of these anomalies [10–25]
was important insofar as it made clear that it is
not straightforward to understand the nonlocality
Accepted in Quantum 2023-11-27, click title to verify. Published under CC-BY 4.0. 6
of boxes and the entanglement of states as two
manifestations of a single type of resource. The
aim of our article is to show that one can
nonetheless do so by recasting the central questions
into a formal resource-theoretic framework. In
particular, we show that if one quantifies the
entanglement properties of quantum states via
LOSR operations, rather than LOCC operations,
then the entanglement of states and the nonlocality
of boxes are indeed seen to be two manifestations of
a single type of resource, namely, nonclassicality of
correlational properties.
We begin by reframing the anomaly of nonlocality
within a rigorous resource-theoretic framework. We
use the framework developed in Ref. [33].6By
viewing entanglement through this lens, we show
that each instance of the anomaly of nonlocality
can be recast as a set of claims that are not
merely counterintuitive but contradictory, thereby
signaling a flaw in the conceptual scheme of
unformalized resource-theoretic assumptions within
which they arose.7
To begin, we note that it is possible to find a
nonlocal box Bwhich can be realized from some
partially entangled pure state of a given Schmidt
rank,
|ψpartial⟩ 7→B, (1)
but which cannot be realized from any maximally
entangled pure state of the same Schmidt rank,8
|ψmax⟩ 7→B. (2)
6
Recall that a set of free operations defines an ordering
relation (formally, a preorder) on resources, where one resource
is at least as resourceful as a second if it can be freely converted
to the second. Two resources are equivalently resourceful (or
in the same equivalence class) if each can be freely converted
into the other, and two resources are incomparable if neither
is freely convertible to the other.
7
Note that earlier work on the anomaly of nonlocality
did not conceive of it as a paradox that was in need of
resolution. The fact that the anomaly becomes a paradox when
recast in a resource-theoretic framework helps us to identify
how to achieve a unified treatment of nonlocality of boxes
and entanglement of states as resources of nonclassicality of
correlational properties.
8
Formally,
|ψmax⟩
is any state for which the squared
Schmidt coefficients describe a uniform distribution for the
given Schmidt rank, while
|ψpartial⟩
is any state for which they
describe a nonuniform distribution.
The following list provides a number of concrete
examples of this phenomenon. For each example, we
specify the box Bappearing therein by reference to
a convex function that witnesses its nonlocality. For
each of the following boxes, one can find a |ψmax⟩
and a |ψpartial⟩of the same Schmidt rank such that
Eqs. (1)and (2)hold:
•a box that achieves the maximum probability of
running Hardy’s version of Bell’s theorem [35].
•a box that maximally violates a tilted Bell
inequality [13,36,37], thereby offering more
noise resistance for that inequality [36].
•a box that has extractable secret key rate higher
than ≈0.144 [38,39].
•a box that has Kullback-Leibler divergence (i.e.,
relative entropy distance) from the set of local
boxes larger than ≈0.058 [40].
Meanwhile, standard entanglement theory tells
us that any partially entangled pure state can be
realized starting from a maximally entangled state
of the same Schmidt rank [41]:
|ψmax⟩ 7→|ψpartial⟩.(3)
It is now evident what is puzzling about these
three claims (Eqs. (1),(2)and (3)): if the conversion
relations in Eqs. (3)and (1)hold in a resource theory,
then given that resource conversion relations are
necessarily transitive in any such theory9—i.e., if
R17→ R2and R27→ R3then R17→ R3—it follows
that we should have |ψmax ⟩ 7→ B, which contradicts
Eq. (2).
We now identify the flaw in the unformalized
resource-theoretic assumptions that led to this
contradiction. It is the implicit idea that the three
conversion relations all hold relative to a single
notion of resourcefulness, that is, that they all hold
relative to the same set of free operations and thus
can be considered as relations holding in one and
the same resource theory. In the description of the
anomaly, the claim about state-to-state conversion,
Eq. (3), is implicitly evaluated relative to LOCC,
while the claims about state-to-box conversions,
9
Transitivity of resource conversions is necessary in the
framework of Ref. [
33
] because the free operations are required
to be closed under sequential composition.
Accepted in Quantum 2023-11-27, click title to verify. Published under CC-BY 4.0. 7
Eqs. (1)and (2), are implicitly evaluated relative to
LO.
In Section 2, we argued that the most
straightforward way of understanding the
interplay of entanglement and nonlocality resource-
theoretically is to imagine a network with a common
source among the parties but no channels, in
which case both state-to-state and state-to-box
conversions should be evaluated relative to LOSR
rather than LO or LOCC. As we now show, this
resolves the contradiction. The standard claims
about the state-to-box conversions are not modified
when one replaces LO by LOSR. Eq. (1)holds
with respect to LOSR because LO ⊂LOSR, and
Eq. (2)holds with respect to LOSR by Lemma 1
and the fact that, for each of the examples given, the
box in question is convexly-extremal in the set of
quantumly realizable boxes. (This follows from the
fact that the functions which the boxes maximize
in each example are convex-linear.) On the other
hand, the standard claim about the state-to-state
conversion is modified when one replaces LOCC by
LOSR. If one judges conversion between entangled
states relative to LOSR, rather than LOCC, then it
is the negation of Eq. (3)that holds, namely,
|ψmax⟩ 7→|ψpartial⟩.(4)
This is because |ψmax⟩and |ψpartial⟩are
incomparable in the resource theory of LOSR-
entanglement—neither can be converted into the
other under LOSR, as shown below (see Corollary 9).
But Eq. (4), unlike Eq. (3), is consistent with Eqs. (2)
and (1), and therefore there is no contradiction (and
hence no anomaly).
The terms “partially entangled” and “maximally
entangled” are apt descriptions of |ψpartial ⟩and
|ψmax⟩when one is considering their LOCC-
entanglement properties. This is because |ψpartial⟩
is strictly below |ψmax⟩in the LOCC order (since,
in addition to Eq. (3), we have |ψpartial ⟩ 7→
|ψmax⟩) and consequently there exists some
LOCC-entanglement monotone, MLOCC, for which
MLOCC
(
ψpartial
)
< MLOCC
(
ψmax
)
and no LOCC-
entanglement monotones relative to which this
strict inequality is reversed. When considering
their LOSR-entanglement properties, however, the
terminology is no longer appropriate. In accordance
with Eq. (4), there necessarily exists an LOSR-
entanglement monotone, MLOSR, relative to which
MLOSR
(
ψpartial
)
> MLOSR
(
ψmax
)
. From this
perspective, it is natural, rather than anomalous,
that there exist tasks—such as realizing the sorts
of nonlocal boxes that appear in the list presented
earlier—for which the type of LOSR-entanglement
required to realize the task is present in |ψpartial⟩but
not in |ψmax⟩. Indeed, one can define a nontrivial
LOSR-entanglement monotone (i.e., one that is
not also an LOCC-entanglement monotone) from
each example of an anomaly of nonlocality. Given
a function over boxes that witnesses the type of
nonlocality described in the example, the LOSR
monotone over states is simply the maximum value
of that function among boxes that are LOSR-
realizable starting from the given state. We provide
the details in Appendix A.
The best known of the anomalies of nonlocality
is the one concerning Hardy’s version of Bell’s
theorem, so it is useful to reiterate our conclusion
for it specifically. The fact that the Hardy-type
correlations cannot be achieved by a maximally
entangled state but can be achieved by a
partially entangled state surprises almost everyone
who encounters the topic. Presumably this is
because—based on their familiarity with LOCC-
entanglement—they expect that whatever resource
of nonclassicality is present in a partially entangled
state, it ought to be less than the resource of
nonclassicality that is present in a maximally
entangled state. The resolution of the puzzle is that
the notion of nonclassicality that is relevant for
Bell scenarios is LOSR-entanglement, not LOCC-
entanglement, and that there are measures of LOSR-
entanglement relative to which what we call a
partially entangled state is more nonclassical than
what we call a maximally entangled state.
The LOSR-incomparability of |ψmax⟩and
|ψpartial⟩also harmonizes with the recently
demonstrated [7] LOSR-incomparability of a
Tsirelson box (which provides the maximal possible
quantum violation of the Clauser-Horne-Shimony-
Holt inequality [42]), denoted by BTsir, and a Hardy
box (which achieves the maximum probability of
running Hardy’s version of Bell’s theorem), denoted
by BHardy. Indeed, both instances of incomparability
can be inferred directly from: (i) the transitivity
of resource conversions within an LOSR resource
theory incorporating both states and boxes, and
(ii) known facts about the possible and impossible
state-to-box conversions under LOSR, namely, that
|ψmax⟩ 7→ BTsir while |ψmax⟩ 7→ BHardy , and that
Accepted in Quantum 2023-11-27, click title to verify. Published under CC-BY 4.0. 8
|ψpartial⟩ 7→ BHardy while |ψpartial ⟩ 7→ BTsir.10 For
instance, to see that these state-to-box conversion
relations imply that |ψmax⟩ 7→|ψpartial⟩, it suffices to
note that if it were the case that |ψmax⟩ 7→ |ψpartial⟩,
then we could follow this conversion with |ψpartial⟩7→
BHardy in order to have a means of converting
|ψmax⟩to BHardy , thereby yielding a contradiction.
Similarly, to see that these relations imply that
BTsir 7→ BHardy, one merely notes that if it were the
case that BTsir 7→ BHardy, then by implementing
|ψmax⟩ 7→ BTsir followed by BTsir 7→ BHardy, one
would have a means of converting |ψmax⟩to BHardy ,
thereby yielding a contradiction. Similar arguments
can be given to establish that |ψpartial⟩ 7→|ψmax ⟩and
BHardy 7→BTsir.11
4 Genuine multipartite entanglement
There is also some tension between results
concerning genuine multipartite entanglement and
those concerning genuine multipartite nonlocality
when the definitions of these concepts are motivated
by the LOCC paradigm for entanglement. We
again begin by reframing this tension as an
outright inconsistency by formulating a genuinely
multipartite anomaly of nonlocality. We consider the
case of three parties for concreteness, although our
analysis can be easily generalized to cases with more
parties.
Denote the entangled state 1
√2(|00⟩+|11⟩)by
|ϕ+⟩. In the preorder of tripartite entangled
states relative to LOCC-convertibility, the state
|ψ2Bell⟩≡|ϕ+⟩A1B⊗|ϕ+⟩A2Cis above the state
|ψGHZ⟩ ≡ 1
√2(|000⟩ABC +|111⟩AB C )because the
former can be deterministically converted to the
latter by LOCC,
|ψ2Bell⟩ 7→|ψGHZ ⟩.(5)
(It suffices for one party to prepare three systems in
their lab in the state |ψGHZ⟩, and then to use |ψ2Bell ⟩
to teleport one part to each of the other two parties,
which requires classical communication.) And yet,
10
These facts about state-to-box conversions are well-known
when the operations are LO, and can be inferred to hold also
for LOSR by appealing to Lemma 1and the convex extremality
of BTsir and BHardy.
11
As an aside, this argument provides an alternative proof
of the LOSR-incomparability of
BTsir
and
BHardy
to the one
presented in Ref. [7].
there are tripartite boxes, such as the Mermin
box [43,44], that can be realized from |ψGHZ⟩by
local measurements,
|ψGHZ⟩ 7→BMermin ,(6)
but that cannot be so realized from |ψ2Bell⟩,
|ψ2Bell⟩ 7→BMermin ,(7)
as follows from a result in Ref. [45].12 As
before, Eqs. (5),(6), and (7)seem to imply
a contradiction given the transitivity of resource
conversion relations.
The resolution of the puzzle proceeds as in the
case of the bipartite anomalies. The reason for
the seeming contradiction is that the conversion
relations of Eqs. (6)and (7)are implicitly evaluated
relative to LO, while that of Eq. (5)is evaluated
relative to LOCC. If, however, one evaluates all
conversion relations relative to LOSR, then although
Eqs. (6)and (7)do not change, by virtue of
Lemma 1and the fact that the Mermin box is
convexly extremal, Eq. (5)does. Specifically, relative
to LOSR, |ψ2Bell⟩and |ψGHZ ⟩are found to be
incomparable, so that the negation of Eq. (5)holds,
that is,
|ψ2Bell⟩ 7→|ψGHZ ⟩,(8)
and the contradiction is blocked. The proof of
incomparability follows from a condition for LOSR-
convertibility we derive further on (see Corollary 8),
as is made explicit in Appendix B.
This anomaly and its resolution shed light on how
one ought to define notions of genuine multipartite
entanglement and nonlocality, and specifically what
it means for these to be genuine 3-way.
We begin by presenting a slightly different
perspective on the anomaly. Consider the
conventional definition of genuine 3-way
entanglement [46,47]. Recalling that a tripartite
state is termed biseparable if there is a partitioning
of the parties into two groups such that the state is
separable relative to this bipartition, the standard
definition can be simply stated as follows:
12
The relevant result from Ref. [
45
] is that the Mermin
box cannot be achieved by LOSR processing from any
state,
ρtriangle
, that is a tensor product of states having
entanglement between pairs of parties only (i.e., generated
from entangled sources consistent with the so-called “triangle
scenario” network),
ρtriangle 7→ BMermin
. Eq.
(7)
follows
because |ψ2Bell⟩is an instance of such a state.
Accepted in Quantum 2023-11-27, click title to verify. Published under CC-BY 4.0. 9
Definition 1 (standard).A tripartite state is
genuine 3-way entangled if and only if it is not a
mixture of biseparable states.
Now note that |ψ2Bell⟩counts as genuine 3-way
entangled by this definition. This is somewhat
counterintuitive a priori, given that |ψ2Bell⟩is
composed of states that contain only bipartite
entanglement. One can glean some insight into what
led to the adoption of the conventional definition, in
spite of its counterintutive features, from the fact
that |ψ2Bell⟩is above |ψGHZ⟩in the LOCC order
(Eq. (5)). Because it is generally agreed that any
reasonable definition of genuine 3-way entanglement
should be such that |ψGHZ⟩counts as genuine 3-way
entangled, it follows that because |ψ2Bell⟩is above
|ψGHZ⟩in the order, it too must qualify as having
such entanglement.
With the conventional notion of genuine 3-way
entanglement in mind, we are now in a position to
present the alternative perspective on the genuinely
tripartite anomaly of nonlocality. First, note that it
is generally agreed that any definition of genuine 3-
way nonlocality should be such that BMermin counts
as genuine 3-way nonlocal. But because |ψ2Bell⟩
is above |ψGHZ⟩in the LOCC order, one would
expect that whatever resource of genuine 3-way
entanglement is required to generate the genuine 3-
way nonlocality inherent in BMermin ,|ψ2Bell⟩would
have it if |ψGHZ⟩does. And yet this intuitive
conclusion is in conflict with Eq. (7).
The problem is that the conventional notion of
genuine 3-way entanglement is motivated by LOCC.
We therefore propose an alternative definition of
genuine 3-way entanglement, motivated by LOSR.
We also show that there is a corresponding
alternative definition for genuine 3-way nonlocality
which mirrors our alternative definition of genuine 3-
way entanglement. The genuinely tripartite anomaly
is shown to admit of a natural resolution relative
to these two notions (genuine 3-way entanglement
and genuine 3-way nonlocality), demonstrating that
these have a more natural interplay than exists
between the conventional pair of notions.
In the LOSR paradigm, the distinction between
a classical and a quantum resource shared among
some parties is the distinction between sharing
classical randomness and sharing entanglement.
Consequently, the natural manner of defining a
resource of 3-way nonclassicality is to consider
resourcefulness relative to a set of operations
wherein all 2-way nonclassicality is considered
free—that is, where 2-way common sources are
allowed to be quantum—while the 3-way common
source is required to be classical. Thus, we define
genuine 3-way nonclassicality for quantum states
and for nonlocal boxes (as well as for other
types of multipartite processes, such as quantum
measurements, quantum channels, and multi-time
processes) as those that are nonfree relative to
LOSR supplemented by 2-way shared entanglement
(LOSR2WSE).13
Definition 2 (resource-theoretic).A nonlocal box
is genuine 3-way nonlocal and an entangled state is
genuine 3-way entangled if and only if they cannot be
obtained by LOSR2WSE, that is, local operations
together with a source of correlations consisting of
shared entanglement between each pair of parties
and shared randomness among all three.
These notions are illustrated in Figs. 1and 2.
Ref. [45] discusses boxes that are genuine 3-way
nonlocal in this sense, and Ref. [26] discusses states
that are genuine 3-way entangled in this sense. The
causal structure of the free resources associated to
LOSR2WSE has been termed the “quantum triangle
scenario with shared randomness” [45,48].
=
Figure 1: A tripartite entangled state is genuine 3-way
entangled if it cannot be decomposed as shown; that is, if
it cannot be realized using LOSR supplemented by 2-way
shared entanglement (LOSR2WSE). Throughout, double
wires represent quantum systems and single wires represent
classical systems.
It is clear that |ψ2Bell⟩can be realized via
LOSR2WSE, while it can be shown that |ψGHZ⟩
cannot (see Ref. [26]), and so of the two, only |ψGHZ⟩
is genuine 3-way entangled according to Definition 2.
In this approach, therefore, the intuitions we noted
13
One can define genuinely
k
-way nonclassicality among
n
parties (where
k≤n
) in an analogous manner: via LOSR
supplemented by (k−1)-way shared entanglement [26].
Accepted in Quantum 2023-11-27, click title to verify. Published under CC-BY 4.0. 10
=
Figure 2: A tripartite box is genuine 3-way nonlocal if
it cannot be decomposed as shown; that is, if it cannot
be realized using LOSR supplemented by 2-way shared
entanglement (LOSR2WSE).
earlier regarding genuine 3-way entanglement are
vindicated: if a state can be obtained from one where
all of the entanglement is of the bipartite variety,
then it is not genuine 3-way entangled.
It is also the case that BMermin cannot be realized
by LOSR2WSE, as follows from results in Ref. [45],
so that BMermin is genuine 3-way nonlocal according
to Definition 2(just as it was relative to the old
definition).
In the description of the tripartite anomaly, the
fact that |ψGHZ⟩can be converted into BMermin
(Eq. (6)) while |ψ2Bell⟩cannot (Eq. (7)) was
only surprising relative to the belief that |ψ2Bell⟩
must have more genuine 3-way nonclassicality than
|ψGHZ⟩does, on the grounds that it is above |ψGHZ⟩
in the LOCC order.14 But this is overturned in the
approach just described, since |ψ2Bell⟩explicitly has
less genuine 3-way nonclassicality than |ψGHZ⟩does,
since |ψ2Bell⟩has none while |ψGHZ⟩has some. Given
that BMermin has genuine 3-way nonclassicality
according to our definition, it becomes intuitively
clear why |ψGHZ⟩and not |ψ2Bell⟩can be converted
into BMermin.
Furthermore, we note that whereas our definition
of genuine 3-way nonclassicality fits within the
mathematical framework for resource theories [33]—
as the property of being nonfree relative to
LOSR2WSE—the conventional definition does not.
The latter fact is most easily seen by recalling
an awkward feature of the conventional definition,
namely, that the set of states that are not genuinely
14
This is in precise analogy to how the fact that
|ψpartial⟩
can be converted to a box manifesting Hardy-type correlations
(Eq.
(1)
) while
|ψmax⟩
cannot (Eq.
(2)
) is only surprising
relative to the belief that
|ψmax⟩
has more nonclassicality than
|ψpartial⟩does.
multipartite entangled is not closed under tensor
products. For example, consider the tripartite
states |ϕ+⟩A1B⊗ |
0
⟩Cand |ϕ+⟩A2C⊗ |
0
⟩B. Both
are biseparable (relative to different partitions
of the tripartite system). Jointly having these
states, however, is equivalent to having |ψ2Bell⟩
=
|ϕ+⟩A1B⊗ |ϕ+⟩A2C, which, as noted previously,
is not biseparable and therefore is genuine 3-way
entangled according to the conventional definition.
Although this feature of the conventional
definition has been acknowledged by some as
counterintuitive and somewhat perverse [26–28],
we wish to draw attention here to the fact that
it is inconsistent with the framework of resource
theories15, because the latter stipulates that the
set of free resources must be closed under parallel
composition (see Definition 2.1 of Ref. [33]). From
the resource-theoretic perspective, therefore, the
fact that the property of biseparability is not
preserved under parallel composition implies that
biseparability is simply not a viable candidate for
the property that defines the set of free resources in
a resource theory. Hence, nonbiseparability is not a
viable candidate for the property that defines the
resource of genuinely multipartite nonclassicality.
Although we have here been primarily concerned
with making the case that it is the notion of genuine
3-way entanglement based on LOSR2WSE, rather
than the one based on LOCC, that does best justice
to our intuitions about multipartite Bell scenarios,
we have also seen that the conventional notion fails
to satisfy certain desiderata that one would want
any resource to satisfy. This raises the question
of whether it even makes sense, within the LOCC
paradigm, to speak of a resource of entanglement
that is genuine 3-way. The following considerations
suggest that it does not.
We begin by highlighting why such a notion
does make sense within the LOSR paradigm of
entanglement. There, the resource of entanglement
constitutes quantumness of common sources.
Consequently, the fact that there is a distinction
between quantum common sources between each
pair of parties and a quantum common source
between all three parties implies that there is a
distinction between 2-way and 3-way notions of
15
Just as the conversion relations in each of the anomalies
were inconsistent with the transitivity of resource conversion
relations.
Accepted in Quantum 2023-11-27, click title to verify. Published under CC-BY 4.0. 11
entanglement. In particular, if all 2-way quantum
common sources are taken to be freely available,
there is still something that the parties can lack,
namely, 3-way quantum common sources.
In the LOCC paradigm, by contrast, the resource
of entanglement is equivalent to a resource of
quantum communication and there is simply
nothing corresponding to the notion of quantum
communication between all the parties beyond the
notion of quantum communication between any pair
of parties. In other words, if all 2-way quantum
communication channels are taken to be freely
available, there is no communication resource that
the triple of parties lack. Such considerations suggest
that it may not be sensible to try and define a
notion of genuine 3-way entanglement in the LOCC
paradigm.
5 Self-testing of entangled states
In this section, we show that LOSR-entanglement is
the notion that best captures many existing ideas
regarding self-testing, i.e., the certification of the
presence of particular entangled states by observing
a nonlocal box [29–32]. 16 More precisely, we show
that many results on self-testing can be rederived
in a methodologically sound manner within the
resource theory based on LOSR, and that simple
generalizations of these results (including some
correctives) follow directly.
Colloquially, a state is said to be self-tested
by a box if this state is “the only one” from
which the box can be obtained by implementing
local measurements. However, there is never just
a single state that can yield a given box, and so
the standard definition allows for some freedom in
the specification of the state. The freedom that is
implied by the original definition [29,30,52,55] is
this:
Definition 3 (standard).The pure state
|ψ⟩
is self-
tested by a box
B
if
B
can be obtained from
|ψ⟩
by
16
Note that it is common to refer to a box as self-testing both
a state and measurements, rather than a state alone [
31
,
49
,
50
].
We will discuss this distinction further in Appendix C. Note
also that we are here considering only the notion of perfect
self-testing, rather than the notion of robust self-testing [
31
,
32
,
51
,
52
], which we do not discuss in this article. Similarly,
we do not consider self-testing based on observing a maximal
violation of a Bell inequality, rather than based on observing
a particular box [50,53,54].
local measurements and if for any state
σ
from which
B
can be obtained by local measurements, there is
a local isometry that takes
σ
to
|ψ⟩⟨ψ|⊗ω
for some
state
ω
. A state
|ψ⟩
is said to be self-testable when
there exists a box
B
such that
|ψ⟩
is self-tested by
B
.
In Appendix C, we discuss a subtlety regarding
the claim that this is the ‘standard’ definition
of self-testing of states. Specifically, we note that
various authors [31,56] did not explicitly require the
condition that Bcan be obtained from |ψ⟩by local
measurements, and we explain why we deem it likely
that these authors were assuming it implicitly.
As we did with the anomalies of nonlocality, we
will begin by taking a resource-theoretic perspective
on self-testing, where the resource theory is based on
LOSR. We take the natural definition of self-testing
to be this:
Definition 4 (resource-theoretic, LOSR-based).
Given a pair consisting of a density operator
ρ
and a
box
B
, we say that
ρ
is self-tested by
B
if it holds that
ρ7→B
and that for all density operators σ,
if σ7→Bthen σ7→ ρ,
where all conversions are evaluated relative to LOSR.
A state
ρ
is said to be self-testable if there exists some
box Bthat self-tests ρ.
In Appendix D, we give an equivalent version
of this definition in terms of the notion of the
upward closure of a resource, that is, the set of all
resources that are above the given resource in the
preorder. Specifically, ρis self-tested by Bif the
upward closure of Bamong states contains only
those states which are also in the upward closure
of ρ. Our definition of self-testing also generalizes
immediately to other sorts of objects and even to
other resource theories, as we discuss in Appendix D.
Note that all notions of conversion appearing in
standard discussions of self-testing (e.g., the notion
of ‘can be obtained from’ in Definition 3) are
understood, within a resource-theoretic approach,
as conversion relative to the free operations in
the resource theory. If self-testing is to be viewed
as the task of certifying the nonclassicality of
correlations in a quantum state by the observation
of the nonclassicality of correlations in a box, then
it follows from the arguments in Section 2that
the appropriate set of free operations is given by